| L(s) = 1 | + 1.43·2-s − 1.25·3-s + 0.0485·4-s − 1.79·6-s + 2.72·7-s − 2.79·8-s − 1.42·9-s + 6.42·11-s − 0.0609·12-s + 4.50·13-s + 3.90·14-s − 4.09·16-s − 7.34·17-s − 2.04·18-s − 2.46·19-s − 3.41·21-s + 9.19·22-s + 6.02·23-s + 3.50·24-s + 6.45·26-s + 5.55·27-s + 0.132·28-s + 0.229·29-s + 4.08·31-s − 0.274·32-s − 8.05·33-s − 10.5·34-s + ⋯ |
| L(s) = 1 | + 1.01·2-s − 0.724·3-s + 0.0242·4-s − 0.733·6-s + 1.03·7-s − 0.987·8-s − 0.475·9-s + 1.93·11-s − 0.0175·12-s + 1.24·13-s + 1.04·14-s − 1.02·16-s − 1.78·17-s − 0.481·18-s − 0.565·19-s − 0.746·21-s + 1.95·22-s + 1.25·23-s + 0.715·24-s + 1.26·26-s + 1.06·27-s + 0.0250·28-s + 0.0426·29-s + 0.733·31-s − 0.0485·32-s − 1.40·33-s − 1.80·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.658776113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.658776113\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 1.43T + 2T^{2} \) |
| 3 | \( 1 + 1.25T + 3T^{2} \) |
| 7 | \( 1 - 2.72T + 7T^{2} \) |
| 11 | \( 1 - 6.42T + 11T^{2} \) |
| 13 | \( 1 - 4.50T + 13T^{2} \) |
| 17 | \( 1 + 7.34T + 17T^{2} \) |
| 19 | \( 1 + 2.46T + 19T^{2} \) |
| 23 | \( 1 - 6.02T + 23T^{2} \) |
| 29 | \( 1 - 0.229T + 29T^{2} \) |
| 31 | \( 1 - 4.08T + 31T^{2} \) |
| 37 | \( 1 - 0.420T + 37T^{2} \) |
| 41 | \( 1 - 5.97T + 41T^{2} \) |
| 43 | \( 1 + 6.55T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 - 2.49T + 53T^{2} \) |
| 59 | \( 1 - 8.13T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 8.55T + 67T^{2} \) |
| 71 | \( 1 - 3.64T + 71T^{2} \) |
| 73 | \( 1 + 8.50T + 73T^{2} \) |
| 79 | \( 1 + 3.50T + 79T^{2} \) |
| 83 | \( 1 + 4.47T + 83T^{2} \) |
| 89 | \( 1 + 4.39T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.328292783307802178372813474455, −6.84300852309118982100136109207, −6.47629169586915051320551105276, −5.95516970306855941751461059403, −5.04239364920152868216932001615, −4.49880035755648445817023806634, −3.96409322397157159664861819980, −3.05494692207258808829855683640, −1.84022298656952030125647175146, −0.793294186550556334433654634955,
0.793294186550556334433654634955, 1.84022298656952030125647175146, 3.05494692207258808829855683640, 3.96409322397157159664861819980, 4.49880035755648445817023806634, 5.04239364920152868216932001615, 5.95516970306855941751461059403, 6.47629169586915051320551105276, 6.84300852309118982100136109207, 8.328292783307802178372813474455
